Well, as for your thread title, I've found that for singletrack type trails, I think rigid is fun. If I'm doing any drops, I'd prefer a suspension fork. You will put in a little more work to ride the same trail on a rigid than a suspension fork, it's just more challenging and fun in some respects.

As for your question about that particular bike, if it's this one, then I'd just make sure everything works as is, give it a tune-up, probably some new tires if the ones it has aren't suitable, and ride it as is. You don't have much invested, so if you don't invest much more then any decent riding bike that fits you should be basically worth it. However, if you see any older bike shop quality MTBs for sale in your area for under ~$75 that are in riding condition(probably with tune-up/tires), I think you'd be better off with that.

This is all assuming by "climb the mountain" you're referring to dirt and/or rocky paths, not paved paths. If they're paved, no need for much other than a hybrid/road bike. Stick some road type tires on the Cross Country and have at it as long as it fits you well. Make sure your seat is high enough that you get proper leg extension.